Tropes are devices and conventions that a writer can reasonably rely on as being present in the audience members' minds and expectations. On the whole, tropes are not clichés. The word clichéd means "stereotyped and trite." In other words, dull and uninteresting.
A trope is like a song in a familiar key; you may not know all the words, but you can probably sing along with the chorus. A cliché, on the other hand, is hoary and old. Not only can you sing along, you know all the words, and you probably can guess the last verse. Sometimes a cliché can be comforting and familiar, but more often it's tired and predictable—and almost certainly predictable and dull to a reader. (There are also linguistic cliches, like "easy as pie" or "sick as a dog." About.com has a good primer on these.)
Let's look at two characters who can shed the light on the difference.
Uncle Iroh of Avatar: The Last Airbender personifies several of my favorite tropes; he's a Badass Grandpa and Old Master who Minored in Ass Kicking.
These tropes, however, are part of a fully formed character; the tropes Iroh embody come from his history, his choices and his own tastes and personality. From the loss that changed the course of his life to his love of tea, Iroh feels real to us because his actions are considered as part of his character—he's much more than a collection of traits assembled into a name and face.
In contrast, the lead character of Ensign Sue Must Die is just a string of clichés put together to serve the plot. Why does she call McCoy by his first name? Who knows? Why is she demanding the attention of every male officer on the Enterprise—is she insecure, an egomaniac? How'd she end up on the bridge at seventeen? Ensign Sue Must Die is a parody, so answering these questions and coming up with a fully formed character isn't particularly important for the creator's purposes. But I'm sure, as a reader, you've found fics that did expect you to sympathize with a character that turned out to be nothing but a collection of personality traits with no inner life or source for that character's motivations. Ensign Sue feels like a cliche—and is—not just because her actions are predictable, but because they come from what the plot requires, not just what she might want or need as a character.
This fantastic blog post on cliché has a great YouTube video of Martin Amis, which includes this quote: Cliché is herd thinking, herd writing.
So much of writing is balance. You want to set yourself apart from the herd-- but not so much that your audience is lost. Tropes, when used well, can do just that. A cliché keeps you hidden in the crowd.
It's a cliché, but in original fiction as well as fanfiction, there's no substitute for hard work. Thinking carefully about your characters, their situation, and your own observations can help keep your sentences and plots free from cliché. (And never underestimate the power of a good beta!)
An additional note
Another reason clichés can be troublesome is the speed at which a cliché can fall into Unfortunate Implications at best, and outright stereotyping at worst.
The Characters as Device page on TV Tropes is long, but quite comprehensive.
Clichés to avoid when writing a mystery story can also be applied to other genres.
Clichés to avoid for romance writers.